Let \(\mathbf {y}_n = [y_1,\ldots ,y_n]^\top \)  denote the vector of noisy observations of \(f\)  at \(\Theta = [\theta _1,\ldots ,\theta _n ]^\top \) , and \(\mathcal {D}_n = \lbrace (\theta _i, y_i), ~i=1,\ldots ,n\rbrace \) , then, applying Bayes' rule, the posterior distribution [1]} is given by
\(\begin{split}f(\theta )|\mathcal {D}_n \sim & \mathcal {GP}(\mu _n(\theta ), \sigma _n^2(\theta )),\\\mu _n(\theta ) =& \mathbf {k}^\top \mathbf {K}^{-1} (\mathbf {y}_n - \mu (\Theta )) \\\sigma _n^2(\theta ) =& k(\theta , \theta ) - \mathbf {k}^\top \mathbf {K}^{-1} \mathbf {k}.\end{split}\) 
